The Number Pi

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From Wikipedia:

“π (sometimes written pi) is a mathematical constant that is the ratio of any circle’s circumference to its diameter. π is approximately equal to 3.14 in the usual decimal notation. Many formulae in mathematics, science, and engineering involve π, which makes it one of the most important mathematical constants.For instance, the area of a circle is equal to π times the square of the radius of the circle.

π is an irrational number, which means that its value cannot be expressed exactly as a fraction having integers in both the numerator and denominator. Consequently, its decimal representation never ends and never repeats. π is also a transcendental number, which implies, among other things, that no finite sequence of algebraic operations on integers (powers, roots, sums, etc.) can render its value; proving this fact was a significant mathematical achievement of the 19th century.

Irrationality and transcendence

π is an irrational number, meaning that it cannot be written as the ratio of two integers. π is also a transcendental number, meaning that there is no polynomial with rational coefficients for which π is a root.An important consequence of the transcendence of π is the fact that it is not constructible. Because the coordinates of all points that can be constructed with compass and straightedge are constructible numbers, it is impossible to square the circle: that is, it is impossible to construct, using compass and straightedge alone, a square whose area is equal to the area of a given circle.This is historically significant, for squaring a circle is one of the easily understood elementary geometry problems left to us from antiquity. Many amateurs in modern times have attempted to solve each of these problems, and their efforts are sometimes ingenious, but in this case, doomed to failure: a fact not always understood by the amateur involved. “